Matrix Groups and Symmetry
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The document discusses matrix groups and symmetry, focusing on linear algebra results such as linear transformations and invertible matrices. It highlights properties of determinants and Euclidean inner products, particularly within the orthogonal group, where the determinant of orthogonal matrices is either 1 or -1. The document also features a theorem relating to the equivalence of certain properties of n x n matrices.
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Matrix Groups and Symmetry
- 1. Matrix Groups and Symmetry Lauren Patterson
- 2. Linear Algebra ResultsLinear TransformationsInvertible MatricesProperties of determinantsProperties of Euclidean inner products
- 6. Example
- 7. Determinant of Orthogonal MatricesThe determinant of any orthogonal matrix is 1 or -1.
- 8. Theorem 10.2Let A be an n x n matrix. The following statements are equivalent:
- 9. Symmetry